ITCS 6114 Graph Algorithms 1 9142021 DepthFirst Search

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ITCS 6114 Graph Algorithms 1 9/14/2021

ITCS 6114 Graph Algorithms 1 9/14/2021

Depth-First Search ● Depth-first search is another strategy for exploring a graph ■ Explore

Depth-First Search ● Depth-first search is another strategy for exploring a graph ■ Explore “deeper” in the graph whenever possible ■ Edges are explored out of the most recently discovered vertex v that still has unexplored edges ■ When all of v’s edges have been explored, backtrack to the vertex from which v was discovered 2 9/14/2021

Depth-First Search ● Vertices initially colored white ● Then colored gray when discovered ●

Depth-First Search ● Vertices initially colored white ● Then colored gray when discovered ● Then black when finished 3 9/14/2021

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d =

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; DFS(G) { for each vertex u G->V { u->color = WHITE; } time = 0; for each vertex u G->V { if (u->color == WHITE) DFS_Visit(u); } } for each v u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } 4 9/14/2021

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d =

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; DFS(G) { for each vertex u G->V { u->color = WHITE; } time = 0; for each vertex u G->V { if (u->color == WHITE) DFS_Visit(u); } } for each v u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } What does u->d represent? 5 9/14/2021

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d =

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; DFS(G) { for each vertex u G->V { u->color = WHITE; } time = 0; for each vertex u G->V { if (u->color == WHITE) DFS_Visit(u); } } for each v u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } What does u->f represent? 6 9/14/2021

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d =

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; DFS(G) { for each vertex u G->V { u->color = WHITE; } time = 0; for each vertex u G->V { if (u->color == WHITE) DFS_Visit(u); } } for each v u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } Will all vertices eventually be colored black? 7 9/14/2021

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d =

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; DFS(G) { for each vertex u G->V { u->color = WHITE; } time = 0; for each vertex u G->V { if (u->color == WHITE) DFS_Visit(u); } } for each v u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } What will be the running time? 8 9/14/2021

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d =

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; DFS(G) { for each vertex u G->V { u->color = WHITE; } time = 0; for each vertex u G->V { if (u->color == WHITE) DFS_Visit(u); } } for each v u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } Running time: O(n 2) because call DFS_Visit on each vertex, and the loop over Adj[] can run as many as |V| times 9 9/14/2021

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d =

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; DFS(G) { for each vertex u G->V { u->color = WHITE; } time = 0; for each vertex u G->V { if (u->color == WHITE) DFS_Visit(u); } } for each v u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } BUT, there is actually a tighter bound. How many times will DFS_Visit() actually be called? 10 9/14/2021

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d =

Depth-First Search: The Code DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; DFS(G) { for each vertex u G->V { u->color = WHITE; } time = 0; for each vertex u G->V { if (u->color == WHITE) DFS_Visit(u); } } for each v u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } So, running time of DFS = O(V+E) 11 9/14/2021

Depth-First Sort Analysis ● This running time argument is an informal example of amortized

Depth-First Sort Analysis ● This running time argument is an informal example of amortized analysis ■ “Charge” the exploration of edge to the edge: ○ Each loop in DFS_Visit can be attributed to an edge in the graph ○ Runs once/edge if directed graph, twice if undirected ○ Thus loop will run in O(E) time, algorithm O(V+E) u Considered linear for graph, b/c adj list requires O(V+E) storage ■ Important to be comfortable with this kind of reasoning and analysis 12 9/14/2021

DFS Example source vertex 13 9/14/2021

DFS Example source vertex 13 9/14/2021

DFS Example source vertex d f 1 | | | | 14 | 9/14/2021

DFS Example source vertex d f 1 | | | | 14 | 9/14/2021

DFS Example source vertex d f 1 | | 2 | | | 15

DFS Example source vertex d f 1 | | 2 | | | 15 | 9/14/2021

DFS Example source vertex d f 1 | | 2 | | | 3

DFS Example source vertex d f 1 | | 2 | | | 3 | | 16 | 9/14/2021

DFS Example source vertex d f 1 | | 2 | | | 3

DFS Example source vertex d f 1 | | 2 | | | 3 | 4 | 17 | 9/14/2021

DFS Example source vertex d f 1 | | 2 | | | 3

DFS Example source vertex d f 1 | | 2 | | | 3 | 4 5 | 18 | 9/14/2021

DFS Example source vertex d f 1 | | 2 | | | 3

DFS Example source vertex d f 1 | | 2 | | | 3 | 4 5 | 6 19 | 9/14/2021

DFS Example source vertex d f 1 | 8 | 2 | 7 |

DFS Example source vertex d f 1 | 8 | 2 | 7 | | 3 | 4 5 | 6 20 | 9/14/2021

DFS Example source vertex d f 1 | 8 | 2 | 7 |

DFS Example source vertex d f 1 | 8 | 2 | 7 | | 3 | 4 5 | 6 21 | 9/14/2021

DFS Example source vertex d f 1 | 8 | 2 | 7 |

DFS Example source vertex d f 1 | 8 | 2 | 7 | 9 | 3 | 4 5 | 6 | What is the structure of the grey vertices? What do they represent? 22 9/14/2021

DFS Example source vertex d f 1 | 8 | 2 | 7 |

DFS Example source vertex d f 1 | 8 | 2 | 7 | 9 |10 3 | 4 5 | 6 23 | 9/14/2021

DFS Example source vertex d f 1 | 8 |11 2 | 7 |

DFS Example source vertex d f 1 | 8 |11 2 | 7 | 9 |10 3 | 4 5 | 6 24 | 9/14/2021

DFS Example source vertex d f 1 |12 8 |11 2 | 7 |

DFS Example source vertex d f 1 |12 8 |11 2 | 7 | 9 |10 3 | 4 5 | 6 25 | 9/14/2021

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13|

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13| 9 |10 3 | 4 5 | 6 26 | 9/14/2021

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13|

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13| 9 |10 3 | 4 5 | 6 27 14| 9/14/2021

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13|

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13| 9 |10 3 | 4 5 | 6 28 14|15 9/14/2021

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13|16

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13|16 9 |10 3 | 4 5 | 6 29 14|15 9/14/2021

DFS: Kinds of edges ● DFS introduces an important distinction among edges in the

DFS: Kinds of edges ● DFS introduces an important distinction among edges in the original graph: ■ Tree edge: encounter new (white) vertex ○ The tree edges form a spanning forest ○ Can tree edges form cycles? Why or why not? 30 9/14/2021

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13|16

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13|16 9 |10 3 | 4 5 | 6 14|15 Tree edges 31 9/14/2021

DFS: Kinds of edges ● DFS introduces an important distinction among edges in the

DFS: Kinds of edges ● DFS introduces an important distinction among edges in the original graph: ■ Tree edge: encounter new (white) vertex ■ Back edge: from descendent to ancestor ○ Encounter a grey vertex (grey to grey) 32 9/14/2021

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13|16

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13|16 9 |10 3 | 4 5 | 6 14|15 Tree edges Back edges 33 9/14/2021

DFS: Kinds of edges ● DFS introduces an important distinction among edges in the

DFS: Kinds of edges ● DFS introduces an important distinction among edges in the original graph: ■ Tree edge: encounter new (white) vertex ■ Back edge: from descendent to ancestor ■ Forward edge: from ancestor to descendent ○ Not a tree edge, though ○ From grey node to black node 34 9/14/2021

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13|16

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13|16 9 |10 3 | 4 5 | 6 14|15 Tree edges Back edges Forward edges 35 9/14/2021

DFS: Kinds of edges ● DFS introduces an important distinction among edges in the

DFS: Kinds of edges ● DFS introduces an important distinction among edges in the original graph: ■ Tree edge: encounter new (white) vertex ■ Back edge: from descendent to ancestor ■ Forward edge: from ancestor to descendent ■ Cross edge: between a tree or subtrees ○ From a grey node to a black node 36 9/14/2021

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13|16

DFS Example source vertex d f 1 |12 8 |11 2 | 7 13|16 9 |10 3 | 4 5 | 6 14|15 Tree edges Back edges Forward edges Cross edges 37 9/14/2021

DFS: Kinds of edges ● DFS introduces an important distinction among edges in the

DFS: Kinds of edges ● DFS introduces an important distinction among edges in the original graph: ■ Tree edge: encounter new (white) vertex ■ Back edge: from descendent to ancestor ■ Forward edge: from ancestor to descendent ■ Cross edge: between a tree or subtrees ● Note: tree & back edges are important; most algorithms don’t distinguish forward & cross 38 9/14/2021

DFS: Kinds Of Edges ● Thm 23. 9: If G is undirected, a DFS

DFS: Kinds Of Edges ● Thm 23. 9: If G is undirected, a DFS produces only tree and back edges ● Proof by contradiction: ■ Assume there’s a forward edge ○ But F? edge must actually be a back edge (why? ) 39 source F? 9/14/2021

DFS: Kinds Of Edges ● Thm 23. 9: If G is undirected, a DFS

DFS: Kinds Of Edges ● Thm 23. 9: If G is undirected, a DFS produces only tree and back edges ● Proof by contradiction: ■ Assume there’s a cross edge ○ But C? edge cannot be cross: ○ must be explored from one of the vertices it connects, becoming a tree vertex, before other vertex is explored ○ So in fact the picture is wrong…both lower tree edges cannot in fact be tree edges 40 source C? 9/14/2021

DFS And Graph Cycles ● Thm: An undirected graph is acyclic iff a DFS

DFS And Graph Cycles ● Thm: An undirected graph is acyclic iff a DFS yields no back edges ■ If acyclic, no back edges (because a back edge implies a cycle ■ If no back edges, acyclic ○ No back edges implies only tree edges (Why? ) ○ Only tree edges implies we have a tree or a forest ○ Which by definition is acyclic ● Thus, can run DFS to find whether a graph has a cycle 41 9/14/2021

DFS And Cycles ● How would you modify the code to detect cycles? DFS_Visit(u)

DFS And Cycles ● How would you modify the code to detect cycles? DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; DFS(G) { for each vertex u G->V { u->color = WHITE; } time = 0; for each vertex u G->V { if (u->color == WHITE) DFS_Visit(u); } } for each v u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } 42 9/14/2021

DFS And Cycles ● What will be the running time? DFS_Visit(u) { u->color =

DFS And Cycles ● What will be the running time? DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; DFS(G) { for each vertex u G->V { u->color = WHITE; } time = 0; for each vertex u G->V { if (u->color == WHITE) DFS_Visit(u); } } for each v u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } 43 9/14/2021

DFS And Cycles ● What will be the running time? ● A: O(V+E) ●

DFS And Cycles ● What will be the running time? ● A: O(V+E) ● We can actually determine if cycles exist in O(V) time: ■ In an undirected acyclic forest, |E| |V| - 1 ■ So count the edges: if ever see |V| distinct edges, must have seen a back edge along the way 44 9/14/2021

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